From Monte Carlo to neural networks approximations of boundary value problems
A talk in the Bielefeld Stochastic Afternoon series by
Iulian Cimpean
| Abstract: | In this talk, we discuss probabilistic and neural network
approximations for solutions to Poisson
equation subject to Holder continuous Dirichlet boundary conditions
in general bounded domains in
Rd. Our main results are two-folded: On the one hand we show that
the solution to Poisson equation
can be numerically approximated in the sup-norm by Monte Carlo
methods, without the curse of high
dimensions and efficiently with respect to the prescribed
approximation error. The proposed approach
reveals that probabilistic representations in conjunction with Monte
Carlo methods are globally efficient
to solve elliptic PDEs, in the sense that the random samples
required by Monte Carlo do not depend on
the location where the solution needs to be approximated. On the
other hand, we show that the obtained
Monte Carlo solver renders ReLU deep neural networks (DNN) solutions
to Poisson problem, whose sizes
depend at most polynomially on d and on the desired error. Within the CRC this talk is associated to the project(s): A5, B1 |